A mechanistic, physically based model
for pollutant release from a surface source, such as field-spread manure,
was hypothesized and laboratory and field-tested. Stable sources and a
conservative "pollutant" (KCl) were used in the laboratory investigation
so that the dynamic effects of source dissolution and chemical transformations
could be ignored and transport processes isolated. The field investigation
utilized runoff and soluble reactive phosphorus data collected from a dairy-manure-spread
field in the Cannonsville watershed in the Catskills region of New York
State. The model predictions corroborated well with observations of runoff
and pollutant delivery in both the laboratory and the field. "Pollutant"
release from surface sources was generally predicted within 11% of laboratory
KCl measurements and field P observations. Laboratory flume runoff predictions
with 15% and 26% errors, for 2.5 cm/hr and 1.5 cm/hr simulated rainfall
intensity experiments respectively, represented root mean square errors
of less than 0.2 ml/s. A 4% error was calculated for overland flow predictions
in the field, which translated into approximately a 0.07 l/s error. Results
suggest that the hypothesized model satisfactorily represents the primary
mechanisms in pollutant release from surface sources.

Introduction

Though runoff from manure spread
fields is identified as a source of nonpoint source pollution, primarily
nutrients, sediment, oxygen demanding compounds, and pathogens (Young,
1976; USEPA, 1990), there are no models which mechanistically describe
transport from a field-spread-manure-type source. The objective of this
project was to create and test a simple pollutant release model which describes
mechanisms for pollutant release from manure-like sources. Furthermore,
in lieu of good understanding of the environmental longevity of important
pollutants, particularly Cryptosporidium parvum, the theoretical
focus was limited to conservative pollutants. A conservative assumption
provides a worst-case scenario. The model was field tested using phosphorus,
which is generally not considered to be a conservative (non-reactive) substance,
to demonstrate that the "conservative" assumption just mentioned doesn’t
seriously impair the utility of this model to substances receiving current
attention in environmental studies.

Agricultural runoff water quality
research has been primarily through field experiments (e.g.: Young and
Mutchler, 1976; Khaleel et al., 1980; Westerman and Overcash, 1980; Edwards
and Daniel, 1993). The modeling efforts aimed at predicting pollutant release
from manured fields have been, by-and-large, empirical and primarily focused
on particulate transport (Wischmeier and Smith, 1978; Khaleel et al., 1979).
Wang et al. (1994) developed a model to simulate runoff transport of land-applied
manure constituents which utilized the convective-dispersion equation and
the Green and Ampt (1911) equation (Mein and Larson, 1973). While this
model is largely mechanistic (Ibrahim and Scott, 1990), it ignores or overly
simplifies the processes around the pollutant source, assuming that rainfall
leaches all soluble constituents from the manure.

To justify limiting the scope of
this work to conservative contaminants, consider one pollutant associated
with field-spread manure which is currently receiving acute attention is
the pathogen, Cyptosporidium parvum (C. Parvum) (Moore and
Zeman, 1991; Anderson and Hall, 1982; Leek and Fayer, 1984; Ongerth and
Stibbs, 1989; Garber et al., 1994; Mawdsley et al., 1995), which has been
identified as the cause of major waterborne outbreaks over the past decade
(Gallaher, 1989; Smith, 1990; Moore et al., 1993; MacKenzie et al., 1994;
Kramer et al., 1995). This organism poses serious problems because it takes
very few individuals to establish infection (DuPont et al., 1995) and expensive
filtration is generally considered the only effective barrier to C.
Parvum. The large quantity of excreted C. Parvum by infected
hosts (Current, 1986) and the robustness of C. Parvum oocysts (Robertson
et al. 1992), including resistance to a wide variety of treatment practices
(Madore et al., 1987; Peeters et al., 1989; Korich et al., 1990; Smith,
1990; Parker et al., 1993; Mayer and Parker, 1996; Chauret et al., 1996),
facilitate their transmission in runoff (West, 1991) and make C. Parvum contamination particularly worrisome. The transport mechanisms of oocysts
in runoff are currently not well understood, research in this area often
showing contradictory results (Brush, 1997). In lieu of better understanding
of entrainment, transport, and survivorship at the microorganism scale,
it is arguable, for the time being, that C. Parvum be treated as
a conservative pollutant in transport modeling.

While the approach used in this project assumes non-reactive contaminants,
its application is not limited to these. The theory presented here is general,
applicable to other substances, and easily modified to account for important
pollutant transformations. Phosphorus (P), which has received particularly
acute attention in recent years, is considered in this study because it
has been well linked with eutrophication and associated ecosystem deterioration
which seriously impairs the value of water bodies as recreation areas and
drinking water sources (Bouldin et al., undated, Sharpley and Smith, 1992)
A major source of P loading to surface waters in the United States Northeast
is dairy farm manure spreading; this problem is exasperated by a trend
towards higher P levels in feed rations over the past 50 years (Klausner
and Bouldin, 1983) and increasing animal intensity (Sharpley et al., 1994).
Better understanding of contaminant transport processes from surface sources,
especially field-spread manure, will ultimately lead to better solutions
for controlling the associated problems.

Model Description

Figure 1 diagrams
the conceptual model. The model simulates three processes: overland flow,
horizontal pollutant convection, and vertical diffusion or convection.
It is assumed that the source height, H, is much greater than the overland
flow depth, h. The depth of flow through the source, hb, is defined as
h/n where n is the source porosity. The source matrix is stable and static.
The possibility of a crust over the source, as is occasionally observed
over dairy manure, is considered in this model; like the source matrix,
the crust is stable and static. At time, t=0, the source is essentially
saturated so that there is no wetting time or change in volumetric storage
within the source.

Overland Flow

Overland flow is modeled with the St. Venant equation:

(1)

Where h is the depth of flow (m), q is the discharge
per width (m2/s), i is the rainfall intensity (m/s), t is time (s), and
x is downhill distance (m). The kinematic assumption applied to the momentum
equation yields (Henderson and Wooding, 1964):

(2)

Where a and m are flow resistance and flow regime parameters respectively. For
turbulent flow these parameters can be approximated with:

(3)

Where n is the Manning roughness
factor (-) and s is the surface slope (m/m).

Using the method of characteristics,
the time from the start of rainfall to reach steady flow, ts, at any point
x is:

(4)

Assuming the time at which the rainfall
ends, tr, is greater than ts, and using the method of characteristics and
equations 1 and 2, the flow at any point, x, can be described by equation
5 through 7.

Rising Limb of Hydrograph (0<t<ts):

(5)

Steady Flow portion of Hydrograph
(ts<t<tr):

(6)

Recession Limb of Hydrograph (tr<t):

(7)

The partial equilibrium situation,
tr<ts, can be analytically described, yielding different expressions
for equations 6 and 7, but is not presented here because the experimental
design did not allow for this situation to be adequately evaluated.

Pollutant Source
Model

The vertical, upper region of the source,
and horizontal, bottom region of the source, pollutant transport mechanisms
are addressed as independent processes. Parameters associated with the
bottom region are designated with a subscript "b". The source, a porous
medium, is divided into two regions dependent on the discharge, q, the
saturated conductivity of the source, Ks, and the bed slope, s. Applying
the Boussinesq equation with the Dupuit assumption, flow in the bottom
region is expressed as:

(8)

Where hb is the depth of flow through
the source (m) (see figure 1) and defines the plane
separating the upper and bottom regions. Invoking the kinematic approximation,
the gradient of the depth with respect to x in equation 8 is approximated
as the ground surface slope, s (negative).

This pollutant release model assumes
two distinct temporal regimes of pollutant release; an early, quick pollutant
release period is dominated by horizontal convection from the bottom region
due to runoff passing through the source and a later, more gradually releasing
period, during which pollutant moves vertically from the upper region.
In the following derivation, much emphasis is given to determining time
tb, the time at which essentially all pollutant is flushed from the bottom
region.

Where c is the concentration of pollutant
(m mole/m3), t is time (s), u is the convective velocity (q/h: m/s), h
is the depth of water flow (n(hb): m), i is the rainfall intensity, and
Jb is the rate of solute uptake from the source into the flow (m mole m-2
s-1). The source porosity is n. The bottom region is flushed of pollutant
at t=tb. As long as pollutant exists in the bottom region, t<tb, and
rainfall has not stopped, t<tr, equation 9 can be solved for concentration,
c, of solution leaving the bottom region using the relationships for h
from equations 2, 5 and 6.

(10)

This solution assumes source matrix
has negligible affect on the mass flow of water, i.e. there is no net storage.
The cumulative mass leaving from the bottom region, Mb, as a function of
time, for t<tb is:

(11)

Where w is the source width (m) perpendicular
to flow. If a crust exists over the source, the mathematical description
of flow at the source is violated, but because the contribution of runoff
near the source is very small relative to flow from a watershed, simply
due to the vastly larger spatial extent of a basin or field relative to
a pollutant source, discontinuities near the source resulting from possible
crusting are negligible. As long as no significant rainfall is stored on
the crust, q can be visualized as the cumulative flow over and through
the source and the concentration is the flow-weighted average concentration
of the contaminated through-flow and clean "over-crust" flow.

The mechanism for rate of pollutant
uptake from the source into the flow, Jb, may be largely dependent on the
characteristics of a particular contaminant, e.g. its dissolving, dissolution,
or desorption rates, and is beyond the scope of this study. As shown later
in this paper, for some pollutants this mechanism is very rapid and its
value can be estimated indirectly without full understanding of uptake
mechanics. Also, the fundamental basis for the model presented here can
be analytically extended to incorporate a larger range of flushing situations;
i.e. it is not mathematically limited to the situation where the period
of rainfall, tr, is greater than tb. Unfortunately, experimental limitations
inhibited investigation of other scenarios.

Vertical Model (upper region):

Pollutant is transferred downward from
the upper region to the bottom region via diffusion when the source is
crusted and via convective-dispersion when there is significant vertical
water flux. Parameters specific to the upper region are denoted with the
subscript "u" and the subscripts "d" for diffusion and "c" for convection
where appropriate.

Because diffusion requires a concentration
gradient, the process is modeled by assuming no diffusive movement until
the bottom region’s concentration is effectively zero, i.e. t³ tb. Assuming the upper region can be approximated as a semi-infinite region,
bounded on the lower end by the top of the bottom region, hb (see Figure
1), where the concentration is zero for t³ tb, Carslaw and Jaeger (1959) showed that the concentration gradient in
the upper region can be written as:

(12)

Where co is the initial concentration,
assumed to be initially homogenous in the upper region (m mole/m3) and
D is the diffusivity (m2/s). Fick’s law can be used to describe diffusive
flux from the upper region, Ju

(13)

Replacing the differential in equation
13 with equation 12 and integrating, the cumulative mass removed from the
upper region by diffusion, Mud, is

(14)

Where A is the source’s horizontal
cross-sectional area (m2), assumed constant for this study.

When water is passing vertically
through the source, convective-dispersion will be the dominant transport
mechanism. The simplest modeling approach is to assume the concentration
of solution released from the source follows a step function.

While
pollutant is available in upper region of the source (15)

When
all available pollutant is removed from the upper region of the source

The cumulative mass leaving the upper
region by convective-dispersion, Muc, is

(16)

Where c is determined by equations
14 and A is the source’s horizontal cross-sectional area (m2). Unlike diffusion,
convective transport is assumed to occur throughout the experiment, not
just after the bottom region is flushed.

Laboratory Experiment

Three conditions were examined in the laboratory: the case with no
crust on the source, the case with complete or full crusting, and an intermediate,
50% crusting, scenario. Each experiment was 30 minutes long and subject
to a rainfall intensity of 2.4 cm/hr. A rectangular sponge, pre-soaked
in a solution of KCl, was used to simulate a pollutant source. The initial
concentration in the sponge was approximately 1 m mole/l; actual values
ranged from 1.0 to 1.3 mmole/l. Figure 2 shows the
flume and source set-up used to test the model. The slope of the flume
is adjustable; for this study s=0.065. Mannings n was taken as 0.05 (Baltzer
and Lai, 1968). Pollutant crusts were simulated using metal covers. The
complete crust was a metal cover over the entire source; the 50% crust
was a metal cover over half the source’s top surface. Diffusivity was taken
as 2x10-5 cm2/s (typical for ionic solutes in water).

Effluent from the flume was collected every 10 seconds for the first
minute, once per minute for the next 29 minutes, and again every 10 seconds
for the recession period after rainfall ceased. Each sampling duration
was 10 seconds. KCl concentration was measured using a pre-calibrated conductivity
meter; care was taken to maintain constant room temperature for all conductivity
measurements to avoid errors arising from conductivity’s thermal sensitivity.
Due to the sponge’s large pores, substantial and prolonged drainage was
observed. Experimental data were adjusted using an empirical drainage curve
developed from laboratory sponge-drainage measurements, for mass removed
via drainage. Before and after the set of experiments simulated rainfall
uniformity was checked using Christian’s coefficient, Cu (James, 1988).
Values for Cu were 0.90 and 0.91 indicating good uniformity. The kinematic
assumptions and parameters were tested during the first two experiments
even though daming the flume with the sponge was expected to attenuate
the rising and falling legs of the hydrograph; results are shown in the
Laboratory Results section.

Xin (1996) demonstrated in similar experiments that the time required
to flush the bottom region of its original pollutants is short, depending
primarily on the time to establish a steady bottom region. By the time
steady state in the bottom region was fully established the region was
essentially flushed. Jb can be approximated as the total mass of salt in
the bottom region per unit area divided by the time to establish steady
flow in the bottom region. It follows then, that tb, the time to flush
pollutants from the bottom region, can be estimated as the time to establish
steady flow in the bottom region:

(17)

Where vb (m3) is the volume of solution in the bottom region of the
source, vo (m3) is volume of water applied to the flume-watershed until
the time steady state overland flow is obtained, ts (equation 4), and v’o
(m3) is the backwater volume. The rate of flow in the system is approximated
as iLw; steady flow conditions implicitly assume ts<<tb. Combining
equations 2 and 4 for h, multiplying by the flume’s width, w (m), and integrating
over the flume’s length, L (m), vo is:

(18)

Where the function ts(L) is the time to steady overland flow at x=L.
With the assumption that the volume of water discharged while t<tb is
small relative to the associated static volumes of vb and v’o, these volumes
can be easily estimated. Due to the flume’s simple geometry, the volume
of the bottom region, vb, is:

(19)

Where H is the total source height (m) (see figure
1) and vt is the total solution volume in the source (m3); measured
values ranged from 300 ml to 450 ml depending on drainage. By continuity,
once steady state is reached, the flow at x=L, q, is iL. Using equation
8 and assuming steady state conditions are valid for this approximation:

(20)

Based on work done by Xin (1996), Ks was taken as 50 cm/min. The
backwater volume is estimated assuming simple daming-up behind the source
of runoff water, i.e. the backwater surface is level. With a constant slope,
s, v’o is estimated by:

(21)

By the definition discussed earlier, the mass transfer flux for the
bottom region is:

(22)

Assuming q is approximated by iL for 0<t<tb (the duration of
the hydrograph rising limb is small compared to flushing period), from
equations 10, 11, and 21:

for t = tb (23)

Laboratory Results

Figures 3 and 4 show the agreement between the kinematic model for
runoff in the flume and experimental data. A statistical comparison between
predicted and observed values is shown in Table 1.
The experiment from which figure 4 was gleaned involved
a "sponge dam" for which this kinematic model does not explicitly apply.

Figure 5 shows the direct cumulative mass
results for the three experiments.

Field observations were made in Delaware County, part of the Catskills
region of New York State. Rainfall, runoff and phosphorus concentration
data were collected from a manure spread field for the June 19 and 20,
1996 rainfall event. Field parameterization was difficult, therefore, order
of magnitude estimates and system simplifications were used to apply the
model. This investigation was to find evidence that the modeled processes
were observable in the field rather than to quantify predictions with great
precision. The runoff contributing area was 2.5 ha which for simplicity
was considered roughly rectangular. The longest flow path length and average
slope were 302 m and 11% respectively. Rainfall intensity was assumed constant
over the event, 0.25 mm/hr. Manure was spread (17.8 T/ha) simultaneously
with the rainfall so source crusting was assumed negligible. Both phosphorus
samples and runoff measurements were automated; phosphorus samples were
taken hourly and runoff discharge measurements were made every 15 minutes.
The samples were tested in the laboratory for soluble phosphorus (P). The
storm commenced before manure application began, therefore the model was
run starting one hour before phosphorus first appeared at the sampling
station (i.e. the last zero phosphorus sample). This starting time happened
to correspond to a slight increase in the hydrograph (Figure
5).

The mathematical development used for the laboratory experiments
was directly applied to the field with the exception of tb determination.
Because of the complicated source geometry in the field a simple empirical
relationship was used to estimate tb based on the laboratory results.

(24)

This relationship assumed similar conductivities between field and
lab pollutant sources. Based on qualitative visual observations, the source
was assumed to be well distributed across the lower portion of the contributing
area in saturated clumps approximately 5 cm high and occupied roughly 1%
of the total runoff-contributing area. Manure conductivity of 50 cm/min
was used. Phosphorus diffusivity was assumed 2x10-5 cm2/s (typical of ionic
solutes in water). At the time of preparing this document, field measurements,
unfortunately, are only available for an uncrusted source. Also, the authors
recognize the that the choice of phosphorus was, in many ways, non-ideal
because of its strong sorption, desorption mechanisms but used it anyway
because of data availability. The field data were used primarily to corroborate
the general modeled mechanisms.

Field Results

The field results are summarized
in figures 4 and 5 and statistical comparisons are shown in Table
3. Figure 6 shows the agreement between the
kinematic model and observed flow. Figure 7 shows
observed and predicted cumulative pollutant release.

Discussion

The kinematic flow equations corroborated
well with the measured data. The oscillating nature of the lab data, seen
in figures 3 and 4, can be largely attributed to the rainfall simulator’s
oscillating mechanism, traveling up and down the flume. The no crust experiment,
which shows the most obvious flow oscillations (squares in figure
4), was run immediately before the 50% crust experiment, which shows
much less dramatic oscillatory characteristics presumably due to damping
associated with the presence of a crust. It is possible, then, that some
of the flow variability is inherent to the experimental apparatus, especially
the rain simulator, which is oscillatory in nature.

The attenuation of the rising hydrograph
limb in the 50% data (circles) in figure 4 show
the "daming" effect of the sponge. The end of the early steep, linear section
of the cumulative pollutant release curves in figure
5 are at t=tb, estimated with equations 16 - 20, and visually correlate
well with the time at which the observed flow (circles in figure
4) reaches steady conditions. This suggests that the approximations,
discussed earlier, in estimating vb and v’o are adequate for this experiment.
The somewhat weaker statistical correlation between the experimental flow
and kinematic equation relative to the high and low intensity experiments
(table 1) may be partially due to the daming period
for which, as discussed earlier, the runoff equations are not designed.

The field data agreed well with the
kinematic equation, perhaps largely due to the fact that preliminary rainfall
had saturated the contributing area making it essentially impermeable;
i.e. flume-like.

The pollutant transport model corroborates
well with the general observed trends in both field and laboratory observations.
The field results show particularly good agreement in that they were much
better than anticipated. The early (t<tb), linear portions of the lab
curves, show good agreement and suggest that the highly simplified nature
of equation 21 is adequate within the boundaries and precision of these
experiments.

The full crust lab experiment showed
the worst statistical agreement between measurements and predictions. While
the full crust experiment corroborated well with predictions during the
initial flushing stage, the total cumulative mass release prediction was
about 20% lower than observed. The discrepancy in total cumulative pollutant
release is due to an under-prediction of the delayed, vertical flux, stage
of release. During this period, the model underpredicted by approximately
30%. This suggests that diffusion is not the only vertical transport mechanism
or that the diffusion coefficient was too low for the very concentrated
solution used in these experiments. One possible explanation is the presence
of vertical convection via rainwater entering the upstream face of the
sponge, which was unprotected (uncrusted). Assuming the exposed upstream
face corresponds to a 10% uncrusted area of the source, and that convected
pollutant from this area can be added to the diffused-out pollutant, corroboration
of predicted pollutant release by measurements improves; R2 is 0.98 and
relative difference is 4.8%. Though the effects of increased solute concentration
on the diffusion coefficient were not investigated, increasing the diffusion
coefficient four-fold makes the prediction-observation agreement similar
to the other two lab experiments; R2 is 0.98 and relative difference is
5.3%. Xin (1996) suggested a dispersion mechanism along the boundary between
the upper and bottom regions to explain the greater-than-diffusion-alone
pollutant release. This explanation may justify a higher "effective" diffusion
coefficient.

In the 0% and 50% crust lab cases
the model’s rate of mass removal for t>tb is much more linear than the
observations (figure 5). This suggests a missing
or inadequately described mechanism in the model, which is present in "sponge"
leaching. Overly linear model results relative to observations are also
present in the field results. One possible explanation is that this model
is the missing dispersion mechanism, which would presumably add curvilinear
characteristics to the predictions. However, the strong statistical agreement
between observed data and predicted results (tables 1 and 2) with the current
simplified model did not warrant additional complication for this study.

This study shows drastic pollutant
release restriction potential for crusted pollutant sources, such as field
spread manure. As shown in figure 5, the crusted
source released only 25-30% as much pollutant at the fully uncrusted source.
For the lab conditions used, increase in pollutant release was roughly
linear with degree of uncrusted area. Using the model to predict fully
crusted pollutant release from the field scenario, predicted differences
in fully crusted and uncrusted pollutant releases were 67%, similar to
laboratory results.

Conclusion

Good corroboration of the model results
with both field and laboratory observations suggests that the model theory
correctly accounts for the primary mechanisms of pollutant transport from
field-spread-manure-type pollutant sources, including potentially crusted
sources. Crusts had an environmentally beneficial affect on pollutant results
in the laboratory. Application of the model to a field scenario suggested
a similar benefits from source crusts in the agricultural environment;
while the lack of data to substantiate this makes the proposal of implementing
manure management strategies that promote crusting premature, these results
suggest the potential for worthwhile research in this area.

Work is needed in accurately parameterizing
field information for the model to expand on model corroboration. Also,
further thought should be given to developing methods to test extended
derivations of the model shown here to account for other situations; for
example, where the pollutant in the bottom region doesn’t flush-out quickly
or the rainfall ends before steady state flow is established. Results could
be somewhat improved with the addition of dispersion to the model, though,
as stated earlier, predictions are fairly good without this complication.
Though further field testing is needed, the results shown above suggest
potential for adaptation of this kind of mechanistic approach to farm management
research and other field situations.